Problem: Multiply the following complex numbers, marked as blue dots on the graph: $( e^{19\pi i / 12}) \cdot (7 e^{23\pi i / 12})$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $ e^{19\pi i / 12}$ ) has angle $\frac{19}{12}\pi$ and radius $1$ The second number ( $7 e^{23\pi i / 12}$ ) has angle $\frac{23}{12}\pi$ and radius $7$ The radius of the result will be $1 \cdot 7$ , which is $7$ The sum of the angles is $\frac{19}{12}\pi + \frac{23}{12}\pi = \frac{7}{2}\pi$ The angle $\frac{7}{2}\pi$ is more than $2 \pi$ . A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $\frac{7}{2}\pi - 2 \pi = \frac{3}{2}\pi$ The radius of the result is $7$ and the angle of the result is $\frac{3}{2}\pi$.